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Developments in Mathematics Developments in Mathematics VOLUME 24 Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California For further volumes: www.springer.com/series/5834 Jerzy Kakol ˛ Wiesław Kubis´ Manuel López-Pellicer Descriptive Topology in Selected Topics of Functional Analysis Jerzy Kakol ˛ Wiesław Kubis´ Faculty of Mathematics and Informatics Institute of Mathematics A. Mickiewicz University Jan Kochanowski University 61-614 Poznan 25-406 Kielce Poland Poland [email protected] and Institute of Mathematics Manuel López-Pellicer Academy of Sciences of the Czech Republic IUMPA 115 67 Praha 1 Universitat Poltècnica de València Czech Republic 46022 Valencia [email protected] Spain and Royal Academy of Sciences 28004 Madrid Spain [email protected] ISSN 1389-2177 ISBN 978-1-4614-0528-3 e-ISBN 978-1-4614-0529-0 DOI 10.1007/978-1-4614-0529-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011936698 Mathematics Subject Classification (2010): 46-02, 54-02 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To our Friend and Teacher Prof. Dr. Manuel Valdivia Preface We invoke (descriptive) topology recently applied to (functional) analysis of infinite-dimensional topological vector spaces, including Fréchet spaces, (LF)- spaces and their duals, Banach spaces C(X) over compact spaces X, and spaces Cp(X), Cc(X) of continuous real-valued functions on a completely regular Haus- dorff space X endowed with pointwise and compact–open topologies, respectively. The (LF)-spaces and duals particularly appear in many fields of functional analysis and its applications: distribution theory, differential equations and complex analysis, to name a few. Our material, much of it in book form for the first time, carries forward the rich legacy of Köthe’s Topologische lineare Räume (1960), Jarchow’s Locally Convex Spaces (1981), Valdivia’s Topics in Locally Convex Spaces (1982), and Pérez Car- reras and Bonet’s Barrelled Locally Convex Spaces (1987). We assume their (stan- dard English) terminology. A topological vector space (tvs) must be Hausdorff and have a real or complex scalar field. A locally convex space (lcs) is a tvs that is lo- cally convex. Engelking’s General Topology (1989) serves as a default reference for general topology. The authors wish to thank Professor B. Cascales, Professor M. Fabian, Professor V. Montesinos, and Professor S. Saxon for their valuable comments and suggestions, which made this material much more readable. The research of J. Kakol ˛ was partially supported by the Ministry of Science and Higher Education, Poland, under grant no. NN201 2740 33. W. Kubis´ was supported in part by grant IAA 100 190 901, by the Institutional Research Plan of the Academy of Sciences of the Czech Republic under grant no. AVOZ 101 905 03, and by an internal research grant from Jan Kochanowski Uni- versity in Kielce, Poland. The research of J. Kakol ˛ and M. López-Pellicer was partially supported by the Spanish Ministry of Science and Innovation, under project no. MTM 2008-01502. Poznan, Poland Jerzy Kakol ˛ Kielce, Poland Wiesław Kubis´ Valencia, Spain Manuel López-Pellicer vii Contents 1 Overview ................................. 1 1.1 General comments and historical facts ............... 7 2 Elementary Facts about Baire and Baire-Type Spaces ........ 13 2.1 Baire spaces and Polish spaces ................... 13 2.2 A characterization of Baire topological vector spaces ........ 18 2.3 Arias de Reyna–Valdivia–Saxon theorem .............. 20 2.4 Locally convex spaces with some Baire-type conditions ...... 24 2.5 Strongly realcompact spaces X and spaces Cc(X) ......... 36 2.6 Pseudocompact spaces, Warner boundedness and spaces Cc(X) .. 46 2.7 Sequential conditions for locally convex Baire-type spaces ..... 56 3 K-analytic and Quasi-Suslin Spaces ................... 63 3.1Elementaryfacts........................... 63 3.2 Resolutions and K-analyticity . ................... 71 3.3 Quasi-(LB)-spaces .......................... 82 3.4 Suslin schemes ............................ 91 3.5 Applications of Suslin schemes to separable metrizable spaces . 93 3.6 Calbrix–Hurewicz theorem . ...................101 4 Web-Compact Spaces and Angelic Theorems .............109 4.1 Angelic lemma and angelicity . ...................109 4.2 Orihuela’s angelic theorem . ...................111 4.3 Web-compact spaces .........................113 4.4 Subspaces of web-compact spaces ..................116 4.5 Angelic duals of spaces C(X) ....................118 4.6 About compactness via distances to function spaces C(K) .....120 5 Strongly Web-Compact Spaces and a Closed Graph Theorem ....137 5.1 Strongly web-compact spaces . ...................137 5.2 Products of strongly web-compact spaces ..............138 5.3 A closed graph theorem for strongly web-compact spaces .....140 ix x Contents 6 Weakly Analytic Spaces .........................143 6.1 A few facts about analytic spaces ..................143 6.2 Christensen’s theorem ........................149 6.3 Subspaces of analytic spaces . ...................155 6.4 Trans-separable topological spaces .................157 6.5 Weakly analytic spaces need not be analytic . .........164 6.6 More about analytic locally convex spaces .............167 6.7 Weakly compact density condition .................168 6.8 More examples of nonseparable weakly analytic tvs ........174 7 K-analytic Baire Spaces .........................183 7.1 Baire tvs with a bounded resolution .................183 7.2 Continuous maps on spaces with resolutions . .........187 8 A Three-Space Property for Analytic Spaces ..............193 8.1AnexampleofCorson........................193 8.2 A positive result and a counterexample ...............196 9 K-analytic and Analytic Spaces Cp(X) .................201 9.1 A theorem of Talagrand for spaces Cp(X) .............201 9.2 Theorems of Christensen and Calbrix for Cp(X) ..........204 9.3 Bounded resolutions for Cp(X) ...................215 9.4 Some examples of K-analytic spaces Cp(X) and Cp(X, E) ....230 9.5 K-analytic spaces Cp(X) over a locally compact group X .....231 ∧ 9.6 K-analytic group Xp ofhomomorphisms..............234 10 Precompact Sets in (LM)-Spaces and Dual Metric Spaces ......239 10.1 The case of (LM)-spaces: elementary approach . .........239 10.2 The case of dual metric spaces: elementary approach ........241 11 Metrizability of Compact Sets in the Class G .............243 11.1 The class G:examples........................243 11.2 Cascales–Orihuela theorem and applications . .........245 12 Weakly Realcompact Locally Convex Spaces ..............251 12.1 Tightness and quasi-Suslin weak duals ...............251 12.2 A Kaplansky-type theorem about tightness .............254 12.3 K-analytic spaces in the class G ...................258 12.4 Every WCG Fréchet space is weakly K-analytic . .........260 12.5 Amir–Lindenstrauss theorem . ...................266 12.6AnexampleofPol..........................271 12.7 More about Banach spaces C(X) over compact scattered X ....276 13 Corson’s Property (C) and Tightness ..................279 13.1 Property (C) and weakly Lindelöf Banach spaces .........279 13.2 The property (C) for Banach spaces C(X) .............284 Contents xi 14 Fréchet–Urysohn Spaces and Groups ..................289 14.1 Fréchet–Urysohn topological spaces ................289 14.2 A few facts about Fréchet–Urysohn topological groups ......291 14.3 Sequentially complete Fréchet–Urysohn spaces are Baire .....296 14.4 Three-space property for Fréchet–Urysohn spaces ........299 14.5 Topological vector spaces with bounded tightness .........302 15 Sequential Properties in the Class G ..................305 15.1 Fréchet–Urysohn spaces are metrizable in the class G ......305 15.2 Sequential (LM)-spaces and the dual metric spaces ........311 − 15.3 (LF )-spaces with the property C3 ................320 16 Tightness and Distinguished Fréchet Spaces ..............327 16.1 A characterization of distinguished spaces .............327 16.2 G-basesandtightness.......................334 16.3 G-bases, bounding, dominating cardinals, and tightness .....338 16.4 More about the Wulbert–Morris space Cc(ω1) ..........349 17 Banach Spaces with Many Projections .................355 17.1 Preliminaries, model-theoretic tools ................355 17.2 Projections from elementary submodels ..............361 17.3 Lindelöf property of weak topologies ...............364 17.4 Separable complementation property ...............365 17.5 Projectional skeletons .......................369 17.6 Norming subspaces induced by a projectional skeleton ......375 17.7 Sigma-products ...........................380 17.8 Markushevich
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